Homoclinic orbits on non-compact Riemannian manifolds for second order Hamiltonian systems
Let \(M\) be a complete connected finite dimensional Riemannian manifold of class \({\mathcal C}^3\). A given function \(V(t,x) \in {\mathcal C}^2 (\mathbb{R} \times M, \mathbb{R})\) determines the second order Hamiltonian system \[ D_t \dot x (t) + \text{grad}_x V \bigl( t, x(t) \bigr) = 0, \tag{1} \] where \(\dot x(t)\) denotes the derivative of \(x(t)\) with respect to \(t\), \(D_t \dot x(t)\) the covariant derivative of \(\dot x(t)\) and \(\text{grad}_x V(t,x)\) the gradient of \(V(t,x)\) with respect to the variable \(x\). If \(x_0 \in M\) is such point that \(V(t,x_0) = 0\), \(\text{grad}_x V(t,x_0) = 0\) for all \(t \in \mathbb{R}\) and \(x(t)\) is a solution of (1) such that \(x(t) \to x_0\), \(\dot x(t) \to 0\) as \(t \to \pm \infty\), we say that there is a homoclinic orbit emanating from \(x_0\). The authors show the existence of a homoclinic orbit on \(M\) under several complex technical conditions on the function \(V(t,x)\) in terms of Riemannian gradient and Riemannian Hessian of \(V(t,x)\). This interesting paper is well written and organized.
scientific article; zbMATH DE number 826763
scientific article